In order to accelerate charged particles to high energies, many types of particle accelerators have been developed since the 1930s. One type of particle accelerator is a cyclotron. A cyclotron accelerates charged particles in an axial magnetic field by applying an alternating voltage to one or more “dees” in a vacuum chamber. The name “dee” is descriptive of the shape of the electrodes in early cyclotrons, although they may not resemble the letter D in some cyclotrons. The spiral path produced by the accelerating particles is normal to the magnetic field. As the particles spiral out, an accelerating electric field is applied at the gap between the dees. The radio frequency (RF) voltage creates an alternating electric field across the gap between the dees. The RF voltage, and thus the field, is synchronized to the orbital period of the charged particles in the magnetic field so that the particles are accelerated by the radio frequency waveform as they repeatedly cross the gap. The energy of the particles increases to an energy level far in excess of the peak voltage of the applied radio frequency (RF) voltage. As the charged particles accelerate, their masses grow due to relativistic effects. Consequently, the acceleration of the particles becomes non-uniform and the particles arrive at the gap asynchronously with the peaks of the applied voltage.
Two types of cyclotrons presently employed, an isochronous cyclotron and a synchrocyclotron, overcome the challenge of increase in relativistic mass of the accelerated particles in different ways. The isochronous cyclotron uses a constant frequency of the voltage with a magnetic field that increases with radius to maintain proper acceleration. The synchrocyclotron uses a decreasing magnetic field with increasing radius and varies the frequency of the accelerating voltage to match the mass increase caused by the relativistic velocity of the charged particles.
In a synchrocyclotron, discrete “bunches” of charged particles are accelerated to the final energy before the cycle is started again. In isochronous cyclotrons, the charged particles can be accelerated continuously, rather than in bunches, allowing higher beam power to be achieved.
In a synchrocyclotron, capable of accelerating a proton, for example, to the energy of 250 MeV, the final velocity of protons is 0.61c, where c is the speed of light, and the increase in mass is 27% above rest mass. The frequency has to decrease by a corresponding amount, in addition to reducing the frequency to account for the radially decreasing magnetic field strength. The frequency's dependence on time will not be linear, and an optimum profile of the function that describes this dependence will depend on a large number of details.